General:
* Connection
* Association
* Bond
* Link
* Interrelation
* Interdependence
* Rapport (especially for a good, harmonious relationship)
Specific to People:
* Affair (for a romantic relationship)
* Friendship
* Partnership
* Union (for a formal union, like marriage)
* Intimacy (for a close, personal relationship)
Specific to Things:
* Correlation (for a statistical relationship)
* Cause and effect
* Interplay
* Dynamic
The best synonym will depend on the specific context in which you are using the word.
For example, if you are talking about the relationship between two countries, "connection" or "association" might be good choices. If you are talking about the relationship between a mother and her child, "bond" or "intimacy" might be more appropriate.